Weak ergodicity breaking and aging in disordered systems

(1)

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Weak ergodicity breaking and aging in disordered systems

J. Bouchaud

To cite this version:

J. Bouchaud. Weak ergodicity breaking and aging in disordered systems. Journal de Physique I, EDP

Sciences, 1992, 2 (9), pp.1705-1713. �10.1051/jp1:1992238�. �jpa-00246652�

(2)

Classification

Physics

Abstracts

75.40 05.40 64.70

Short Communication

Weak ergodicity breaking and aging in disordered systems

J. P. Bouchaud

Service de

Physique

de l'Etat Condensd,

CEA-Saclay,

91191Gif-sur-Yvette Cedex, ^France

(Received

25

May

J992,

accepted

in

final form

J9June J992)

Abstract. We present a

phenomenological

model for the

dynamics

of disordered

(complex)

systems. ^We

postulate

that the lifetimes of the many metastable states are distributed

according

to a broad, _power law

probability

distribution. We show that

aging

_occurs in this model when the average lifetime is infmite. A

simple hypothesis

leads to a new functional fornl for the relaxation which is in remarkable agreement ^with

spin-glass experiments

over

nearly

five decades in time.

In

spite

of fifteen _years of

dispute,

the

theory

of

equilibrium spin-glasses

is not

yet

^settled

[1, 2, 3]. Dynamical

effects _are,

however, likely

to be the dominant aspect ⁱⁿ

experiments.

One of the _most

striking

aspects ^of ^the

dynamics

of

spin-glasses

in their low temperature

phase

is the

aging phenomenon-

a rather

peculiar

and awkward feature from the

thermodynamics point

of view _: the relaxation of _a system

depends

_on its

history.

More

precisely,

^if a system ^is field-cooled below its

spin-glass temperature,

^the

magnetization

relaxation

depends

_on the

Waiting

time t~ ^between ^the

quench

and the switch off of the

magnetic

field

[4, 5, 6].

Similar effects _are observed _on the viscoelastic

properties

of

polymer

melts

[7], magnetic properties

of HTC

superconductors [8]

and _more

recently

_on the

relaxation after _a heat

pulse

in

Charge Density

Wave systems

[9]. Analytical

fits of the

magnetization

relaxation _as _a function of time have been

proposed.

^In

[6],

it is

proposed

^that

the initial

«stationary»

_part of the relaxation is _a

power-law

With _a small

(negative)

exponent. ^For times

longer

than the

Waiting time,

relaxation is Well fitted

by

_a _« stretched _»

exponential decay, provided

that _an effective time is introduced

[6].

In

[4, 5], however,

_a

stretched

exponential

of the «real» time _was found for

relatively

short times.

Many

phenomenological

theories have been devised to account for the stretched

exponential decay lo-14],

and for the slow part ^and

aging [6, 15-18].

We however feel that the basic mechanism

underlying aging

has not been

fully appreciated (see

however

[17]) although

it

is,

in _our

opinion,

_one of the constitutive

properties

of

spin-glasses.

The aim of this _note is _to

suggest

that

aging

is related to

ergodicity breaking

which has, in these systems, a

peculiar

and

perhaps unexpected meaning.

We find _on _some

simple

models that-

according

_to _our

definition,

_see below _« Weak _»

ergodicity breaking

occurs in the

spin-glass phase,

defined

as the

phase

Where the Edwards-Anderson parameter ^is non-zero. We _see however _no _reason

Why

the two should be linked in

general,

and suggest ^that ^the appearance of

aging

could be taken _as _an

operational

definition of the

spin-glass

transition.

(3)

1706 JOURNAL DE PHYSIQUE I N° 9

We

reproduce

both the short time

(t«t~)

and the

long

time

(t»t~) parts

^of the

magnetization decay

under

quite

reasonable

hypotheses.

The

analytical

form of this

decay

is however different from all those

previously proposed.

We find in

particular

that _:

m(t)

₌ _mo _exp

i- y/(I x) (t/t~)~ ~~i

0 _w _x _w

for _t _« t~ ^and a power law

decay

m(t)

=

(tit~)-Y

for _t » t~. ^Note ^that ^the two

regimes

are

strongly connected,

since the above exponent ^is related to the

parameters describing

the initial

part

of the relaxation. This feature should be _a

stringent

test of _our

theory.

As _We shall _see, _our

theory

is in remarkable agreement ^With ^both the _« short _» time

experiments

^of

[4. 5],

and the very

long

time data of

[6].

It is

Widely accepted

that the energy

landscape

of _a finite disordered system ^is

extremely rough,

^With _many ^local ^minima

corresponding

to metastable

configurations,

Which _We shall call

loosely

_« states _». Since the energy

landscape

is

rough,

these local minima _are surrounded

by

rather

high

_energy barriers. We thus

expect

^that ^these states act as « traps » Which

get

^hold of the

system during

_a certain time _r.

What is the distribution of these

trapping

^times

~(r)

? A

simple picture

for the energy

landscape

is the

following (Fig. I,

_see also

[11])

_: there exists _a _«

percolation

_» _energy level

fo

below which

« states _» are disconnected _;

fo

is the minimal energy

required

_to _«

hop

_»

between any ^two ^states. In other

words,

_we _assume that the

dynamics

benveen the traps ^is _very fast and that the

probability

to find the

system

^between two metastable states is

negligible.

r

(

a

r

b)

^AGING

io

r

Fig.

I. a) Schematic view of the energy

landscape

_: holes _are drilled below the reference energy

fo

which is the minimal energy needed to go from _one metastable state to another. Note that this

drawing

is _one dimensional _: in

reality

mountains of

height

»

f~

also exist between different states. b) When the extemal field is cut, the

landscape acquires

_a non-zero

slope

which drives the system towards zero

magnetization

states.

(4)

In this

picture,

the energy barrier AE is thus

equal

to the

depth

of the trap,

fo f.

In the

spin glass phase

^of ^both the Random

Energy

Model

(REM)

and of the SK

model,

the distribution of very low f's is

exponential [1, 23, III

:

P

~f )

= NIT exp x

( f fo)lT (1)

where _x is _a temperature

dependent

number between 0 and

I, fo

^is ^the reference level above Which levels

proliferate

and N is _a _constant. In the

REM,

_x

=

T/T~

^Which ^is ^connected to the fact that the free energy

landscape

is

temperature independent.

This is not the _case in the SK

model,

Where

x(T)

is non-trivial

[I].

Its

shape

is Well

approximated by

^the _«

one-step

_»

replica breaking

scheme

[5].

Assuming

that AE=

fo- f,

and that

r =

roexp(AE/T),

one

finds, using (I)

and

~

(T)

dr

=

P

( f ) df

_:

~(y)

^~yx y- (I +x)

(2)

0

for

large

_r.

[w

^is a normalization

constant].

Such _a form is also obtained in many other

models- for

example

^the one-dimensional «random force model»

II?, 20-22],

with

x =

TF/« where F is the average force and wits variance.

It may be that for real

spin-glasses

the

probability

distribution of low free energy ^states

decays

more

slowly

for

large negative f,

for

example

_as _a _power law

P

(f )

=

fi(- f )~

^~~ ^+'~

(3)

with ( _~ ^0. ^In ^this case, ~

(r)

reads _:

~

(r )

=

air

jog rir~j-

^(' ⁺ ^<)

(4)

where 8 is _a constant. This will lead _to

logarithmic,

_as

opposed

to power

law, decay. (4)

would

probably

conform _more to the ideas of Fisher and Huse

[15].

The most

interesting property

of these distribution laws is the fact that

IT)

w

=

drr~(r) diverges,

when xwl for

equation (2)

and for all finite

(

for

o

equation (4).

Hence,

the time needed _to

explore

_an infinite system îs înfinite. ^This îs ^however a non-

conventional scenario for

ergodicity breaking,

since the

phase

_space is _a

priori

not broken into

mutually

inaccessible

regions

in which local

equilibrium

_may be achieved. We shall call this situation _« weak _»

ergodicity breaking.

It may be that

« true _»

ergodicity breaking

also _occurs, I.e. the existence of many « pure ^states » between which infinite barrier stand. We shall

carefully

avoid this issue

(the

reader is referred to

[19]

for _a _recent

experimental discussion),

since the

dynamics

is

by

definition restricted _to

only

_one _pure state, and for which the concept of _« weak _»

ergodicity breaking

is relevant.

Let _us _now _see how this feature is

deeply

connected _to the

possibility

of

observing aging.

Suppose

that the system ^under

study

can be

decomposed

into ~lL

independent subsystems.

These

subsystems

_may

be,

for

example, weakly coupled polycristalline

_«

grains

_» or clusters of

magnetic

atoms

they

are not, in _our

mind,

_« domains _»

growing

with time _as is assumed in references

[15, 16] (although

such domains may indeed be present inside each

grain).

In other

words,

_we _assume that _an

experimental sample

is

already

_a collection of _a

large

number of micro

spin-glasses,

and hence that the measured

quantities

_are disorder averages. We claim that

performing dynamical experiments

_on

mesoscopic samples,

where ~lL is

small,

would

yield irreproducible

results.

(5)

Each

subsystem

I

(

=

I,

~lL) _can be in _a certain number of _« states _»

8(I )

= 8 which _we shall suppose

roughly independent

of I. The total number of _states is thus

=

8~

Note that in the SK

model,

8

depends

_on

temperature,

^with ⁱⁿ

particular 8(T~)

₌ I.

The system ^is ^field ^cooled to a certain

temperature T~T~.

Since _a

magnetic

field H is _on, the system ^evolves ^within a «

pool

of _states

» which carry a net

magnetization

but

cannot reach states with _a different value of this

magnetization

_: this would _cost _an extensive

(free)

_energy. The free energy distribution of these states is characterized

by

_a certain

x(T, H)

_or

((T, H) depending

_on which of the two distributions

(I)

_or

(3)

is _more

adapted. [For convenience,

_we shall

adopt

in the

following

the distribution

(I),

the

corresponding

results for distribution

(3)

_can be obtained from those

given

below

by defining

a time

dependent

effective parameter x

(t) according

to

x(t

=

(I

₊

~) log(log t/ro)/log(t/ro)

which goes to zero as t _-

oo].

The

magnetic

field is left _on _a time t~.

During

this

time,

the

subsystems

start

exploring

their

phase

_space. The

deepest

trap ^available ^for a

given subsystem

I is characterized

by

a

trapping

time

t(I)

such that _:

s

°~

dr ~

(r)

=1

(5)

t(1)

Equation (5) simply

_means that _out of 8

trials,

_a value of

r

larger

_or

equal

to

t(I )

has been encountered

roughly only

_once.

(A

more

rigorous justification

_can be found in e.g.

[21]). Thus,

from

(5), t(I )

= To

8"~.

_This

longest trapping

^time ^of _course fluctuates from

one

subsystem

to the

other,

with _a distribution

given by £(t, 8)

such that

£

(t, 8)

=

r(

Sit ⁺ ~

(6)

for t »

8~'i [The

maximum relaxation time available in the whole system is

thus, similarly, t(~lL)

= 8_~'~~~.]

Let _us _now estimate the order of

magnitude

of the

longest trapping

time r~~

actually

encountered

during

a time t~, ^when _r~~~ ^is ^much ^smaller ^than

t(I).

If N

(t~)

denotes the number of different _« _states

» visited

by

the

subsystem

after t~, _r~~~ ^is ^obtained

through

_:

N

(tw ) °~

dr ~

(r )

>

(7a)

lr~

t~ ₌

£ _rj

m N

(t~)

dr r~

(r ) (7b)

j i. N(iw) ⁰

which

gives,

for _x

~ l and for all finite (, _r~~

= t~. ^This ^is ^the

key point

in _our

picture

: the

deepest

state encountered

traps

^the system

during

_a time which is

comparable

to the overall

waiting

time.

Schematically,

states such that _r «

t~

are

thus,

for _x _~

l, ergodically probed,

whereas those for which _r » t~ are

barely

visited. This is not so for _x

~ l _: from

(7a, b)

one

finds that the

typical

value of _r~~~ is

r/ "~(t~)~'~

« t~.

We _are _now in _a

position

to describe the observed relaxation laws

quantitatively.

When the

magnetic

field is switched

off,

the free energy of the metastable states are

uniformly

shifted

by

an amount + MH

(Fig. lb).

The

subsystems

will evolve

independently

towards _states

carrying

no

magnetization,

which

requires

a collective

change

of _a finite fraction of the

spins.

The system must leave the metastable state that it had entered when the field _was _on.

However,

the system may also continue _to

probe deeper

and

deeper

wells while

relaxing;

its age t~ ^is ^thus ^time

dependent

with _:

t~(t)

= t~ + t. This feature was

recognized

as a crucial

ingredient

^for a consistent

analysis

^of the

experimental

data

[6].

(6)

Two _cases must then be

distinguished

_:

A)

Short

waiting

times _:

Suppose

first that the

waiting

time t~ ^is ^much ^shorter ^than

Tog "§

defined

by equation (5).

In this

regime,

the

phase

_space is

effectively

infinite. The

probability

P

(r,

_t~

)

to find _a

given subsystem

in _a _state characterized

by

_r _can

easily

be shown to be

equal

to

r(r/t~ ) r~(r),

with

r(u)=

I for small _u, and

decaying

to zero for u»I. The detailed form of

r(u)

would

depend

on the _« geometry » of

phase

_space I.e. the relative

positions

and

interconnections between the different metastable states. In the

simple

_case where all states are

equally accessible,

_one finds

r(u)

= I/u for _u » I.

The very curious feature in the _x

~ l

phase

is that _r

~

_r is _a

priori

not

normalisable,

since

IT )

is infinite. P

(r,

_t~

)

_can thus

only

be normalised when the

waiting

time t~ ^is

finite,

and then reads _:

P

(r,

t~

)

=

Ar

(r/t~ ) (t~

_)~

'/r~ (8)

lw

with A

=

dur(u)/u~ [=

III _-x if _one takes _a

simple step

^for

r(u)].

Note that the

o

microscopic

time _To has

entirely disappeared

from

equation (8)

This would _not be the _case for _x _~ l _: in this _case, _one

simply

finds P

(r,

t~

)

_cc _r ~

(r ),

which in fact

corresponds

to the

ergodic

result

(indeed, r~(r)

is

proportional

to the Boltzmann

weight).

Assuming

_a

simple exponential decay

from

single

states

('),

_we thus

find,

for the

magnetization

relaxation _rate _:

lm

dm/m

=

pi I/rj

dt

= dt

lp/Zj

dr I/r P

(r,

t +

t~) exp(- t/r) (9)

o

lw

where Z

=

dr P

(r,

t + t~ exp

(- t/r).

_p is the

probability

for the

magnetization

of the

o

subsystem

to relax when

leaving

_a _r-trap, rather than continue

aging

_among

magnetized

states. p thus _measures _« how easy » it is to find _states with small

overlaps [24].

Aging during

relaxation is taken into account

by

the appearance oft ₊ t~ ⁱⁿ ^the ^above

expression.

Formula

(9)

is

easily

transformed into _:

dm/m

= t

-~(t

₊ _t~

)~~

^' G

(tit

₊ _t~

)

dt

(10)

where G

(u)

is another

slowly varying function,

related _to

r(u),

such that G

(O)

=

pr(x)/A.

Note that

equation (10)

is close in

spirit, although

not

equivalent,

to the _treatment of references

[6, 7],

where _a

description

of

aging during

relaxation _was

proposed.

In the

following

_we shall

neglect

the variations of

G,

and

study (10)

in the short and

long

time

regimes.

For _t _« t~ one finds

m(t)

₌

moexpi- y/(I -x) (t/t~ )' ~~i

=

mo(I y/(I -x) (t/t~

_)~ ^~~ +

) (II)

where _y

=

pr(x)/A.

For _t » t~,

aging during

relaxation becomes the

predominant

feature and _we find _:

m

(t)

=

(tit~ )-

^Y

(12)

(1) ^Note however that

our central result, equation (10), is

unchanged

_up to a redefinition of G if exp(- t/r) is

replaced by

_any function of t/r decaying faster than t~

(7)

(assuming

that

G(I

₌ G

(O)).

Three remarks must be made _:

I)

_our

theory reproduces

the

ubiquitous

_« stretched _»

exponential decay,

which is _seen _to be

only

valid at short times _t « t~.

One should also _note that based _on the

exponential

distribution of

free-energies (Eq. (I)),

De Dominicis et al.

[I II

have

proposed

a

dynamical

model for which a stretched

exponential decay

is found for

long times,

with _an exponent x

(while

_we find I _-x, _see

Eq. (ll)).

However,

_our

approach

is very different since _we _assume that the

hopping

rate from _one state to another is determined

by

the _«

starting

_state _» and not, as in

[I ii, by

the _« destination _»

state. Furthermore, _no

aging

effects _were found in this model.

ii) (I I)

^and

(12)

show that the _two limits

(short

and

long times)

are

intimately

connected.

iii)

Note that in this

regime,

_m

(t,

t~

)

is _a function of the ratio

t/t~ only.

In

particular,

_we find that the short time

decay,

for t « t~, is not

stationary.

This is _at variance with the

theory

of references

[15, 16],

where the

equilibrium (stationary) dynamics

is

predicted

for

t«t~_ ^In our

model,

this is

only

the _case when t~» _To

8"~ _(see _below).

_Moreover

m(t~,

t~

)

is still of order of mo.

We have fitted _some of the

experiments

of

[6]

with the function obtained from the

integration

of

(10),

with _a _constant G. This leaves three

fitting

parameters: mo, x, and _y.

(Note

however that the

shape

of the _curve is

only parametrized by x).

As _can be _seen from

figure 2,

the agreement îs ^remarkable ⁱⁿ ^view ôf ^the ^broad ^scale ôf times which is

probed

_: t varies from 0.I _to 3 500

min,

and the

waiting

time t~ ^is

equal

to 31.3 min. For T

= 0.6

T~,

we find _x

= 0.76 and _y

= 0.062 _;

using

A

=

I/(I x),

this

corresponds

to p of the order of 1/3. Note in

particular

that _an

algebraic

distribution of lifetimes

(Eq. (3))

_appears to be _more

appropriate

than the

logarithmic distribution, equation (4).

T=10K, x=0.7645, y=0.0645, m(t=0)=0 567

O.6

o.5

O.4

O.3

O.Ol 0.I 10 loo 1000 1001

log(t[min.])

Fig.

2. Fit of _one of the

aging experiment

_on

Cdcrj_~Ino,354 spin glass

[6]. The

waiting

time is 31.3 min under 15 Gauss, and the temperature ^is

10K=0.6T~.

The set of paranJeters ^used was

mo = 0.567 (in units of the field cooled

magnetization),

_x

=

0.76 and _y

=

0.0645.

(8)

Other

temperatures

^and

waiting

times _are

equally

well

reproduced by

_our

theory.

A _more extensive

presentation

^of ^these

^fits,

^and of the very

interesting

field and temperature

dependence

^of ^the parameters, ^will ^be

published

elsewhere

[25].

Note

finally

that the

experiments

of

[5]

have been

analyzed

in terms of _a stretched

exponential decay.

This 15

nicely explained by

our

theory,

since in these

experiments,

the

decay

time is less than _or

comparable

to t~.

In order to

emphasize

the

peculiarity

of the

region

x~

I,

_one should note that for

x~l,

where the distribution

(8)

becomes

normalisable,

_one finds

m(t)=mo

[I (t/ro)

+ : the

magnetisation

has thus

drop substantially

after _a

microscopic

time To, and not after the

macroscopic

time t~.

B) Long waiting

times and

«

interrupted aging

_»

Let _us _now consider the other limit t~ »

Tog

~'§ ^which

corresponds

to the _case where most systems ^have ^reached

equilibrium. Using (6),

_we find that

(8)

is

replaced by

_:

p(y) gyy~l+xj~/gI/~ _j gl-16( )~-l/~x _(j~)

" T0 ~ T0

for _r «

Tog

^~'~

Hence,

for _t _«

Tog ^~'~«

t~, we find that

m

(t)

=

mo(i yi(i x) (t/ro

s

i/x)i

^-x

) (14)

which is

independent

of t~.

For

ro8"~«t«t~,

one observes those

subsystems

such that

t(I)~t~,

for which

equilibrium

is reached. From the

theory

of

L£vy

_sums

(see

_e.g.

[21]),

one may show that the

deepest traps

^of ^these

subsystems,

for _x

~

l, entirely

dominates the

partition

function. In other

words,

the

probability

_to

find

the

subsystem

I in its

deepest

state is

oforder

I. The fraction

m(t)

of

subsystems

still

carrying

their initial

magnetisation

is thus, for

Tog ^"~«

t « t~,

m(t)/mo lm

= dr

(rj8/r~ _+~j _exp(- _t/r)

=

8(ro/t)~

« l

(15)

o

where _we have used the

probability

distribution

(6).

Hence

aging essentially

_stops

(or

becomes

inobservable)

when

Tog

^"~=

t~~~, beyond

^which ^the ^finite ^size ^of ^the

phase

space appears. «

Interrupted aging

_» has indeed been observed in

Charge Density

Waves

[9],

and

can be used _to _measure the _«

complexity

_» n of the system, ^which ^is

naturally

defined _as

n

=

Logs

=x

Log (t~~~/ro).

^Let us

finally

mention that

aging

of low

frequency

_a-c-

susceptibility

_can be understood

using

_an extension of

equation (9)

_:

x

(w, t~)

=

°~

dr p

(r,

t +

t~) i/(i

+

iwr) (16)

where _a

simple Debye

form is assumed for each _state

[25].

In

particular,

we find that

x

"(w, t~)

=

4 (&

= w

t~)

with

4

&

)

= & for small it, and

4

&

)

=

&~ ^' for

large

&i, in agreement ^with

Koper

and Hilhorst

[16].

We have thus

proposed

a

phenomenological

model which allows _us to

understand,

_on _very

general grounds,

how the

phenomenon

of

aging

sets in _: the

phase

_space must posses a

large

number of metastable states, which _a

broad, scaleless,

distribution of lifetimes. When the average lifetime of these metastable states

diverges,

all the

physical

observables _are dominated

by

the

properties

of the

deepest

_state that the

system

was allowed to

probe

in its

(9)

quest ^for ^the « true »

equilibrium

state.

Aging

_stops either when the distribution of lifetimes is truncated and is _a finite size

effect,

_or when the distribution of lifetimes

decays

more

rapidly

than

r~~

_A

more formal

description

of _our results could be

given, along

^the lines of reference

[11], using

_a Master

equation

in

phase

_space.

The most

important quantity

of _our

theory

^is ^the ^index _x, which

naturally

_appears in the SK

or REM models of

spin-glasses.

_x

gives

information _on the distribution of

free-energies

in the

spin-glass phase,

and the present ^model ¹⁵ a

suggestion

for its

experimental determination,

which would be of great ^interest. ^The ^behaviour ^of x and the

disappearance

of

aging

_as

T~

^is

approached

determines the nature of the

freezing

_at the

glass point

_: in the Random

Energy

Model

[23], x(T~)

=

I,

which _means that the

phase

_space is broken into many

metastable states of

comparable

_« size _». In the SK

model,

_on the other

hand, x(T~)

₌

(one valley dominance) [II

and the number of states

S(T-T~)

^is ^tends to I _:

aging

is then

interrupted

for

microscopic

times.

Let us

finally emphasize

that _our model is not restricted to

spin-glasses

and may thus be useful to understand

aging

in other systems ^such as

polymer glasses [7]

and

Charge density

waves

[9].

Acknowledgements

I _want to thank Eric Vincent with whom I have had many crucial

discussions,

in

particular

_on

the

experimental

data. This paper has been

strongly

influenced

by

these discussions.

Important

remarks and

claryfing

comments are also due to M.

Feigelman,

J.

Hammann,

F. Le

Floch,

M. Ocio in

Saclay

and P.

Cizeau,

B.

Derrida,

D. S.

Fisher,

A.

Georges

and

M. M£zard.

Finally,

_an

interesting

discussion with R. Orbach must be

acknowledged.

Part of this work has been

performed

_at Laboratoire de

Physique Statistique,

Ecole Normale

Sup£rieure.

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large

times.

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